Dozenal Directions

Background: Prior Art
Cardinal Directions
Older directional names appear for the four cardinal directions originating from a North and South wind axis and a dawn to dusk axis.

The Twelve Classical Wind Directions
Elaboration and refinement of the cardinal directions with intercardinal directions produced eight directions, then extended to twelve by philosophers. Classically in Greek and later in Latin, the twelve directions were named by winds blowing from those directions. These were:

Wind source direction	Greek	Latin
North pole	Aparktias	Septentrio
Eastern Arctic circle	Meses	Aquilo
Summer solstice sunrise	Kaikias	Caecias
Equinox sunrise	Apeliotes	Subsolus
Winter solstice sunrise	Euronotoi	Eurus
Eastern Antarctic circle	Phoinikias	Euroauster
South pole	Notos	Auster
Western Antarctic circle	Libonotos	Austroafricus
Winter solstice sunset	Lips	Africus
Equinox sunset	Zephyros	Favonius
Summer solstice sunset	Argestes	Corus
Western Arctic circle	Thraskias	Circius

Adaptation of these wind directions was proposed for the orienteering directions as of magnetic compasses or to be shown on maps. However, while the resulting angles of these directions may be approximately consistent with dozenal division of the circle at the latitudes in which the Mediterranean civilisations sailed, the directions of the rising and setting points at the horizon of a celestial body such as the sun on the ecliptic vary with latitude.

Binary division of directions
More recent compasses from medieval times have been based on successive bisections of the angles between cardinal directions, as for example on portolan maps. The names of the resulting eight principal wind directions were:

Direction	Latinate name
North	Tramontana
Northeast	Greco
East	Levante
Southeast	Scirocco
South	Ostro
Southwest	Libeccio
West	Ponente
Northwest	Maestro

Compass directions by explorers from Iberia may have been influenced by Islamic navigators.

Astronomically derived angular measure of direction
By rising and setting points on the horizon
It was shown in the table above of the twelve classical wind directions how the solar rising and setting points on the horizon could have been used to determine intercardinal directions.

The principle of celestial rising and setting directions could be extended from the sun, which changes during the year, to the fixed stars, which have the same rising and setting points all year round. However, the directions of rising and setting of the fixed stars still vary with latitude and furthermore their declinations alter over generations due to precession, causing the celestial systems which were once used to be now obsolete.

I found it surprising that the binary compass directions were assigned by the Muslims to fixed stars roughly according to their declinations, even though declination is a polar angle, measured between the pole and equator, whereas the compass direction is an azimuthal angle, measured along the horizon of the observer at a particular latitude. It seems to be an excessively inaccurate and complicated system when simply a protractor and the single pole star at night, without having to wait for a star to rise or set, or bisection of equally long shadows of the sun on opposite sides of the noon meridian during the day could perform the same function even without a magnetic compass, except in the case of obscuring weather when no method of observing stars would work. In any case, by the method of following stars, many more of the same declination would need to be known in order for a navigator to maintain a course when the assigned one is far above the horizon.

Another way by which angular measurement for direction could arise is from measurement of time.
From division of the year: degrees and sexagesimal angles
From the approximate number of days in the year, the Babylonians divided a circle into three hundred and sixty degrees. It is possible to encounter modern compasses that are graduated with the three hundred and sixty degrees of the circle, with emphasis on the twelve directions at sequential intervals of thirty degrees. In modern astronomy, minutes and seconds of arc are current and in principle could be used for relative direction.

From division of the day: hours
In ancient Egypt, sundials were used in which dawn to dusk was divided into twelve hours. In principle, if the sundial was set level, then the hour marks would point to directions of orientation, though not necessarily at equal angular increments.

Twelve hours on an ancient Egyptian sundial. Image source: https://en.wikipedia.org/wiki/Sundial#/media/File:Ancient-egyptian-sundial.jpg

The Chinese used two dozen named directions on compasses, which came about from an astronomical system.

Two dozen Chinese compass directions on a Korean sundial. Image source: https://en.wikipedia.org/wiki/Sundial#/media/File:휴대용_앙부일구.jpg

Dozenal directions proposal
The following proposal is to divide the circle of directions only by powers of twelve, first into twelve sectors and then each of the twelve sectors further into twelve, to produce twelve squared directions in total. Divisions into twelve are to be achieved by alternation of division by four and then three. Thus, the first subdivision produces the four cardinal directions and quadrants. The second subdivision trisects these quadrants. These two subdivisions produce the division into twelve sectors.

The method of naming the directions in English is as follows:

1. The cardinal directions are the same, namely East, North, West, and South.
   
2. The directions resulting from the trisection are named from previous directions by their combination in the form x of y, where y is the angularly closer direction and x is the angularly farther direction, and such that the compound direction is a third of the way from Y to X. The abbreviation for the combination is XoY, where X and Y are the initial letters of the cardinal directions being compounded.
   
3. In the second quartering of directions, those directions that bisect the cardinal directions are named by compounding the names of those cardinal directions in the same way as is customary in the binary scheme.

Table of Dozenal Compass Directions

Angle from East in radians				Direction name	Abbreviation
Quarters	Twelfths	Half twelfths	Quarter Twelfths
0*tau/4				East	E
			tau/④⓪	East Northeast of East	ENEoE
		tau/②⓪		Northeast of East	NEoE
			3*tau/④⓪	East Northeast	ENE
	tau/①⓪			North of East	NoE
			5*tau/④⓪	East Northeast of Northeast	ENEoNE
		3*tau/②⓪		Northeast	NE
			7*tau/④⓪	North Northeast of Northeast	NNEoNE
	2*tau/①⓪			East of North	EoN
			9*tau/④⓪	North Northeast	NNE
		5*tau/②⓪		Northeast of North	NEoN
			⑪*tau/④⓪	North Northeast of North	NNEoN
tau/4				North	N
			①①*tau/④⓪	North Northwest of North	NNWoN
		7*tau/②⓪		Northwest of North	NWoN
			①③*tau/④⓪	North Northwest	NNW
	4*tau/①⓪			West of North	WoN
			①⑤*tau/④⓪	North Northwest of Northwest	NNWoNW
		9*tau/②⓪		Northwest	NW
			①⑦*tau/④⓪	West Northwest of Northwest	WNWoNW
	5*tau/①⓪			North of West	NoW
			①⑨*tau/④⓪	West Northwest	WNW
		⑪*tau/②⓪		Northwest of West	NWoW
			①⑪*tau/④⓪	West Northwest of West	WNWoW
2*tau/4				West	W
			②①*tau/④⓪	West Southwest of West	WSWoW
		①①*tau/②⓪		Southwest of West	SWoW
			②③*tau/④⓪	West Southwest	WSW
	7*tau/①⓪			South of West	SoW
			②⑤*tau/④⓪	West Southwest of Southwest	WSWoSW
		①③*tau/②⓪		Southwest	SW
			②⑦*tau/④⓪	South Southwest of Southwest	SSWoSW
	8*tau/①⓪			West of South	WoS
			②⑨*tau/④⓪	South Southwest	SSW
		①⑤*tau/②⓪		Southwest of South	SWoS
			②⑪*tau/④⓪	South Southwest of South	SSWoS
3*tau/4				South	S
			③①*tau/④⓪	South Southeast of South	SSEoS
		①⑦*tau/②⓪		Southeast of South	SEoS
			③③*tau/④⓪	South Southeast	SSE
	⑩*tau/①⓪			East of South	EoS
			③⑤*tau/④⓪	South Southeast of Southeast	SSEoSE
		①⑨*tau/②⓪		Southeast	SE
			③⑦*tau/④⓪	East Southeast of Southeast	ESEoSE
	⑪*tau/①⓪			South of East	SoE
			③⑨*tau/④⓪	East Southeast	ESE
		①⑪*tau/②⓪		Southeast of East	SEoE
			③⑪*tau/④⓪	East Southeast of East	ESEoE

Square twelfths are omitted for brevity and expedience. Their names and abbreviations are formed by compounding such that the word "of" appears twice.

Advantages of this proposal include that

• it maintains the current binary compound directions unchanged whereby
  
• bisections are simple compounds of the bisected directions.

References

• https://en.wikipedia.org/wiki/Cardinal_direction
  
• https://en.wikipedia.org/wiki/Classical_compass_winds
  
• https://en.wikipedia.org/wiki/Compass_rose
  
• https://en.wikipedia.org/wiki/Points_of_the_compass
  
• https://en.wikipedia.org/wiki/Sundial

See also:

• https://dozenal.forumotion.com/t38-air-current-zones
  
• https://dozenal.forumotion.com/t16-angular-to-rectangular
  

The "Dozenal directions proposal" outlined above on Monday the 21st day of September last year was entirely my own idea not seen by me anywhere else before at that time of writing. It is based on the manner in which I have been giving directions verbally in public for years. However, some weeks ago this year 2021, I found substantially the same idea, though less developed, in posts of another author on another forum website. I had searched that website before to check what the proposals there on the same topic had been, but did not see this idea there at that time. Therefore, I believe that the other author has edited his past posts, after begging for permission from the forum administrators to do so, in order to make it look as though this idea of mine was his own because it was superior to his ideas on the matter. I have given credit to this other author before where it was due, but in this case I think he is simply passing off my idea as his own, or at least someone else has done this for him. It must be very hard for the other author that nearly all of the best ideas in dozenism have come from someone other than himself. One person has devised the best dozenal symbological system, the best dozenal nomenclatures, and the best dozenal metrological proposal, and many other optimum dozenal proposals, but it is not him.